LOOPS OF EPICYCLOID AL CURVES. 



125 



9. When one of the radii, say OA, becomes indefinitely large the epicycloid 

 merges into the cycloid produced by carrying the centre of a revolving wheel 

 along a straight line ; and the extension of Perigal's problem leads naturally to 

 this one : — 



" To construct a cycloid of which the loops may touch each other." 



10. If we suppose the centre of the revolving circle to be carried along the 

 axis Y with a linear velocity v, while the radius B turns with an angular velocity 

 (3, the co-ordinates of the tracing-point are 



x = B . cos {3l ; y = vt + B . sin f3t 



and for the point of contact we must have 



= vT + B . sin /5T 

 = v + /3B . cos jST 



wherefore all such points are determined by the solution of the trigonometrical 



equation 



/3T=tan/3T; 



that is to say, we must discover all those arcs which are equal in length to their 

 own tangents. 



11. If we put v = (3=l, we obtain the following solutions for the first ten 

 cases, the first of these being that of the common or cusped cycloid. 



T 



Log sec T 



B 











00 



00-00 



0-000 0000 



1-000 00 



257 



27 



12-24 



0-663 0732 



4-603 34 



442 



37 



27-57 



0-891 5209 



7-789 70 



624 



45 



36-54 



1039 4093 



10-949 88 



805 



56 



00-77 



1-149 2717 



14-101 71 



986 



40 



35-75 



1-237 7832 



17-289 54 



1167 



11 



22-88 



1-309 5424 



20-395 88 



1347 



33 



55-30 



1-371 8187 



23540 67 



1527 



51 



08-52 



1-426 2642 



26-684 82 



1708 



04 



43-65 



1-474 6296 



29-828 38 



VOL. XXIV. PART I. 



2m 



